Detection by multiple trellises

726 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 3, MARCH 2009

Detection by Multiple TrellisesM. Franceschini, Member, IEEE, G. Ferrari, Member, IEEE, and R. Raheli, Member, IEEE

Abstract—In this paper, we present a novel pragmatic ap-proach, referred to as detection by multiple trellises, to performtrellis-based detection over realistic channels. More precisely,we consider channels with unknown parameters and apply theconcept of detection by multiple trellises to forward-backward(FB) algorithms. The key idea of our approach consists, first,of properly quantizing the channel parameters and, then, con-sidering replication of coherent FB algorithms operating onparallel trellises, one per hypothetical quantized value. In orderto make the receiver robust against a possibly time-varyingchannel parameters, the proposed soft-output algorithms per-form a proper “manipulation” of the forward and backwardmetrics computed by the parallel FB algorithms at regularlyspaced trellis steps. We consider two significant examples ofapplication: detection over (i) phase-uncertain channels and (ii)fading channels. The performance of the proposed algorithms isinvestigated considering differentially encoded (DE) quaternaryphase shift keying (QPSK) and iterative detection schemes basedon low-density parity-check (LDPC) codes. Besides having alow complexity, the proposed soft-output algorithms turn outto be robust, flexible, blind, in the sense that no knowledgeof the channel parameter statistics is required, and highlyparallelizable, as it is desirable in high-throughput future wirelesscommunication systems.

Index Terms—Forward backwards algorithm, non-coherentdetection, soft-input soft-output detection, LDPC codes.

I. INTRODUCTION

DETECTION over channels which depend on unknownparameters, i.e., detection with unknown channel state

information (CSI), has long been an active research fieldin the literature. Several signal processing techniques havebeen developed in the last decades to overcome possibleimpairments of the communication channels. Since the in-troduction of turbo-codes (TC) more than a decade ago [1],a great effort has been devoted to develop soft-input soft-output (SISO) detection algorithms suitable for iterative pro-cessing [2]. While SISO algorithms were first derived forthe additive white Gaussian noise (AWGN) channel, theyhave been extended to more realistic channels, such as thoseof interest in wireless communications. In particular, thesechannels are often characterized by time-varying parameters,

Paper approved by H. Leib, the Editor for Communication and InformationTheory of the IEEE Communications Society. Manuscript received April 28,2007; revised August 14, 2007.

M. Franceschini was with the Department of Information Engineering,University of Parma, Italy. He is now with the IBM T. J. Watson ResearchCenter, 1101 Kitchawan Road, Yorktown Heights, NY 10598 (e-mail: [email protected]).

G. Ferrari and R. Raheli are with the Department of Information Engi-neering, University of Parma, viale G. P. Usberti 181A, I-43100 Parma, Italy(e-mail: {gianluigi.ferrari, raheli}@unipr.it).

This paper was presented in part at the IEEE Intern. Symp. Inform. Theory(ISIT’05), Adelaide, Australia, Sept. 2005, and the Intern. Symp. of WirelessCommun. Syst. 2005 (ISWCS 2005), Siena, Italy, Sept. 2005.

Digital Object Identifier 10.1109/TCOMM.2009.03.070185

either stochastic or deterministic but unknown. In addition, thestatistics of the stochastic parameters may not be availableat the receiver. An example of such channels is the phase-uncertain channel [3], where the transmitted signal undergoesan unknown phase rotation and is affected by AWGN. Anotherrelevant example is a fading channel [4], which arises becauseof unresolvable multipath in radio communications.

Two main approaches to perform detection over channelswith parametric uncertainty can be devised:

• separate detection and parameter estimation [3];• joint detection and parameter estimation. In the case

of phase-uncertain communications, parameter estimationmay be embedded in the detection process, explicitly [5]or implicitly [6]–[9].

In [10]–[14], linear predictive receivers for fading channelsare proposed, considering the Clarke model for fading chan-nels [15], [16]: these receivers exploit the correlation charac-teristics of the fading process to predict its evolution. Anothergeneral approach consists of describing the evolution of thefading process through a suitable Markov chain [17]–[19], andthen taking this model into account in the receiver design [20]–[24]. A major issue, in the design of SISO algorithms forrealistic channels, is to obtain good performance together withhigh robustness against channel variations and low computa-tional complexity.

In this paper, we present a class of low-complexity SISOalgorithms, derived from the standard forward-backward (FB)algorithm1 [25]. These algorithms stem from an optimalapproach to detection for channels affected by block-constanttime-varying unknown parameters. In particular, after a properquantization of the channel parameters, the proposed algo-rithms consist of running a number of coherent standard FBalgorithms in parallel, which exchange information only at arelatively small number of fixed trellis epochs. As relevantcase studies, we focus on differentially encoded (DE) qua-ternary phase shift keying (QPSK) over (i) phase-uncertainand (ii) flat fading channels. We focus our attention on theperformance of the proposed algorithms in the low signal-to-noise ratio (SNR) region, which is of interest in modern com-munication systems. As a consequence, our analysis focuses,besides on a simple DE-QPSK scheme, also on schemes basedon the concatenation of a low-density parity-check (LDPC)code [26] with DE-QPSK. In both phase-uncertain and fad-ing channels, the obtained results show that the proposedalgorithms have good robustness and low sensitivity to thestatistics of the channel parameters. In the fading scenario,we compare the performance of the proposed approach withthat obtained by modeling the fading process by a first-order

1Also widely known as BCJR algorithm from the initials of the originalproposers [25].

0090-6778/09$25.00 c© 2009 IEEE

Authorized licensed use limited to: Universita degli Studi di Parma. Downloaded on March 19, 2009 at 09:43 from IEEE Xplore. Restrictions apply.

FRANCESCHINI et al.: DETECTION BY MULTIPLE TRELLISES 727

Markov-chain [17]–[19]. Finally, we show that by applyingthe principle of detection by multiple trellises, it is possible toobtain low-complexity soft-output algorithms with negligibleimpact on the system performance. In addition, the proposedalgorithms do not rely on any statistical information on thechannel parameters (either phase or fading), i.e., they areblind. We note that a multiple-trellis approach was previouslyproposed to cope with the problem of joint detection and mod-ulation classification in [27], where the unknown parameter(modulation format) was inherently static.

This paper is organized as follows. In Section II, weintroduce the idea of detection by multiple trellises. In Sec-tion III, we apply the multi-trellis SISO algorithms to DEphase-uncertain communications and analyze, by computersimulations, the performance of the proposed algorithms. InSection IV, we extend the derivation proposed in the previoussection to flat fading communications and compare the resultswith a Markov chain-based approach. A simple complexityanalysis is presented in Section V, comparing the proposedalgorithm with known solutions. Section VI concludes thepaper. In the Appendix, a formulation of an FB algorithmfor transmission through a Markov chain channel is given.

II. DETECTION BY MULTIPLE TRELLISES: THE IDEA

In order to set the problem under study and present themathematical notation, we begin by reviewing a modifiedversion of the FB algorithm suitable for generic finite-memorychannels affected by time-invariant stochastic parameters.Afterwards, we will describe the extension to time-varyingparameters and propose two different multi-trellis SISO algo-rithms.

A. Time-Invariant Parameters

Let us assume that the channel output is observed fora period of K + 1 symbol intervals. The channel can becompletely described by the following joint probability densityfunction (PDF)

p(rK0 , ξ|aK

0 ) (1)

where rK0 is the vector of the observables (r0, . . . , rK),

ξ ∈ Dξ is a stochastic channel parameter independent of thetransmitted data, Dξ is the domain of the channel parameter,and aK

0 is the vector of information symbols ak transmittedthrough this channel. Note that (1) can take into accountpossible coding of the information symbol sequence {ak} intoa code sequence {ck}. We remark that the parameter ξ couldbe either a scalar parameter or a vector parameter, i.e., ξ couldrepresent a whole set of parameters.

The a posteriori probability (APP) of an information sym-bol ak can be expressed as follows:

P{ak|rK0 } ∝ p(rK

0 |ak)P{ak}

= P{ak}∫Dξ

p(rK0 |ak, ξ)p(ξ) dξ (2)

where the notation “∝” indicates that the first member isproportional to the second through a constant independent ofthe transmitted information symbol ak. If, conditionally on theparameter realization ξ, the channel has finite memory [9],

the conditional PDF p(rK0 |ak, ξ) can be computed via a

standard FB algorithm [2], [25]. This is possible wheneverthe transmission system can be modeled as a finite statemachine (FSM) whose input and output are, respectively, theinformation symbol ak and a random variable (RV) whosestatistics depend only on the FSM state and the input symbol.

A simple approximation for the computation of the integralin (2) consists of performing the following finite sum:

P{ak|rK0 } ∼∝ P{ak}

L∑i=1

p(rK

0 |ak, ξ(i))

p(ξ(i)) (3)

where {ξ(1), . . . , ξ(L)} is a set of quantized values for thechannel parameter whose positions and number L are chosento obtain the desired accuracy in the numerical integrationin (2). This corresponds to running L standard FB algorithms(each one associated with a value ξ(i), i = 1, . . . , L) in paralleland computing a weighted average of their outputs to obtaina quantity approximately proportional to the APP.2

In the following, we denote the forward state metricscomputed during the forward recursion of an FB algorithm as{α(i)(sk)}, where the superscript i refers to the FB algorithmassociated with the quantized parameter value ξ(i) and sk

denotes the state of the FSM in the corresponding trellisdiagram. In particular, we assume that sk ∈ {0, . . . , Ξ − 1},where Ξ is the number of states characterizing each trellis.Similarly, we denote the backward state metrics associatedwith the i-th trellis diagram as {β(i)(sk)}.

Several practical scenarios can be cast within the modeldescribed by (1), (2) and (3). In particular, as useful examples,we will consider phase-uncertain and flat fading channels.

1) Phase-Uncertain Channel: In a communication scenariowhere the channel introduces a time-invariant phase rotation,the stochastic channel parameter ξ can be equivalently mod-eled as a phase rotation θ of the transmitted symbol sequence.The discrete-time equivalent observation can be expressed as

rk = ckejθ + nk (4)

where rk is the received observable, ck is the (possibly coded)transmitted symbol, and nk is a (noise) sample of a sequenceof independent and identically distributed (i.i.d.) zero meanGaussian RVs.

2) Flat Fading Channel: The generic observation modelgiven by (1) applies directly to a flat fading channel, providedthat ξ has the proper statistical distribution. In particular, ina scenario with unresolvable multipath, ξ corresponds to afading coefficient f and the channel input-output relation canbe expressed as follows:

rk = f ck + nk (5)

where, in the case of Rayleigh fading, f has a complexcircularly-symmetric Gaussian distribution with zero mean.

B. Time-Varying Parameter

The idea of detection by multiple trellises stems froman extension of the previous static-parameter approach to ascenario with time-varying channel parameters.

2We implicitly assume that the reader is familiar with the FB algorithm.More information can be found in [2], [25].



{

{{

l-th parameter block (l + 1)-th parameter block

associatedwith ξ(1)

Trellis

associatedwith ξ(2)

Trellis

Trellis

associated

lN − 1 lN lN + 1

with ξ(L)

Fig. 1. Time-varying trellis for detection on block-constant discrete parameterchannel.

In order to obtain insights on the impact of the presence ofa time-varying parameter, let us consider a useful case studywhere the channel parameter process {ξk} is discrete andblock constant. Let us assume that ξk is uniformly distributedover the set {ξ(1), . . . , ξ(L)} and constant over blocks of lengthN < K . In other words,

ξlN+i = ξlN+j ∀i, j ∈ {0, . . . , N − 1}

and the realizations {ξk} are i.i.d. from block to block, i.e.,

p(ξlN , ξnN ) = p(ξlN )p(ξnN ) =1L2

∀l �= n .

As a consequence, the process {ξk} is a time-varying Markovchain, characterized by an L×L transition matrix Pk = (p(k)

ij )at the k-th epoch such that

p(k)ij =

{δij if k �= N − 1 mod N1L

if k = N − 1 mod N

where δij denotes the Kronecker delta. We further assumethat the information sequence {ak} is encoded into a codesymbol sequence {ck} by means of an FSM. Consideringthat the observed sequence of length K comprises more thanone length-N block with constant channel parameter, theapplication of a maximum a posteriori (MAP) strategy to thisscenario leads to a time varying trellis. In the Appendix, ageneral formulation of a MAP algorithm for a finite-memorychannel characterized by a generic Markov-chain parameteris presented. In Fig. 1, a representative time-varying trellisfor this illustrative block-constant discrete parameter channelis shown. Within a block, i.e., for N − 1 consecutive timeepochs, the trellis structure consists of L “coherent” trellises,each assuming knowledge of ξ, one for each quantized value ofξ. In the sections of the various trellis diagrams connecting thestates at the end of a block with the states at the beginning ofthe next block, each state in each coherent trellis is connectedwith the corresponding state in all the other coherent trellises.In other words, each coherent trellis is connected with anyother trellis by the non-zero probability of variation of theparameter value.

Applying the general formulation in the Appendix, theforward and backward metrics αk(sk, ξk) and βk(sk, ξk) arefunctions of the “extended” state σk = (sk, ξk). They can be

computed recursively by running L separate coherent FB algo-rithms, one for each parameter value. Every N time steps, ingeneral, αk(sk+1, ξk+1) and βk(sk, ξk) depend on all forwardand backward metrics in all coherent trellises, respectively, i.e.,a “mix” of the forward and backward metrics in the coherentFB algorithms is performed. The above considerations can beequivalently drawn by following the guidelines in [21], wherea Markov-chain model for the channel phase is assumed.

At this point, the idea underlying detection by multipletrellises can be outlined. As for a constant channel parameterξ, several coherent FB algorithms are run independently,characterized by forward and backward metrics α

(i)k (sk) =

αk(sk, ξ(i)) and β(i)k (sk) = βk(sk, ξ(i)), respectively. The

difference with respect to the time-invariant channel parametercase is that every N time steps, the forward (backward)metrics in the different trellises are properly “mixed” toaccount for the possible variation of the channel parameter.In the following, we will refer to N as “inter-mix interval.”

The idea of considering parallel trellises which occasionally“talk” to each other is appealing, since it is likely to allowboth low-complexity and parallel processing. In Section V,we will investigate the complexity of the proposed algorithmsand compare them with other existing solutions. In this sense,performing detection by multiple trellises can be equivalentlyinterpreted as an instance of the divide et impera approach totackle complicated problems with limited complexity.

The “mix strategy,” in general, should be tailored for thespecific communication scenario at hand. Nevertheless, somegeneral considerations can be drawn:

• If ξ is time-invariant, the quantity p(rK0 |ak, ξ(i)), com-

puted via a coherent FB algorithm, is expected to bemaximum in correspondence of the value ξ(i) closestto the true3 channel parameter ξ. In fact, numericalanalysis in several scenarios showed that the forward andbackward state metrics {α(i)

k (sk)} and {β(i)k (sk)} exhibit

an exponential decay4 as a function of the epoch k. Thisis due to the fact that, if α

(i)k is the vector of the forward

metrics at epoch k in the i-th trellis, the forward recursioncan be equivalently expressed as

α(i)k = Γ(i)

k−1α(i)k−1 (6)

where Γ(i)k is a matrix whose elements are the pdfs of the

observable rk conditioned to every possible transitionsin the i-th coherent trellis. In particular, as expected, thedecay exponent is greater in the FB algorithm associatedwith the phase value ξ(i) which is closest to the true chan-nel parameter ξ, leading to state metrics {α(i)

k (sk)} and{β(i)

k (sk)} relatively much larger than those computedby the j-th FB algorithm with j �= i.

• If ξ is time-varying, we expect that {α(i)k (sk)} and

{β(i)k (sk)} will try to adapt to the parameter changes.

3Depending on the symmetry structure of the modulation code, i.e., thelaw encoding the information symbols ak into the transmitted symbols ck,there can be a set of ξ values which are optimal, in the sense that they areundistinguishable at the receiver. This may occur, for example, in differentialM -PSK transmitted over a phase uncertain channel, where phase rotations ofthe observed sequences by multiples of 2π/M cannot be distinguished [8],[21].

4In the probability domain.



This adaptiveness is limited by the fact that state metricsexhibit a “low-pass filter” behavior, i.e., state metricshave memory and can change only slowly. This is dueto the recursive structure of the metric computationalgorithm (6). In other words, the FB metric computationprocess can be equivalently described as a recursive time-varying vector filtering.

• While in standard applications an FB algorithm is in-sensitive to a possible multiplication of all forward orbackward state metrics by a constant, in the algorithmunderlying (3), the relative weights of different trellisesare important. Accordingly, the multi-trellis SISO algo-rithm turns out to be insensitive to a normalization ofthe metrics only if this normalization is carried out, at agiven epoch, over all forward or backward state metricsof all parallel FB algorithms.

In the following, two possible “mix” strategies are proposed.These strategies will be analyzed in Section III and IV.

1) Multi-Trellis SISO Algorithm 1: At each length-N in-terval, i.e., at epochs k = lN , l ∈ N, one could manipulatethe forward metrics {α(i)

k (sk)} (and, similarly, the backwardmetrics {β(i)

k (sk)}) according to the following rule:

α(i)k (sk) ←−

L∑j=1

α(j)k (sk) i = 1, . . . , L ∀sk (7)

where the notation “←−” represents the assignment of anew value. This corresponds to averaging, for every givenstate sk, the metrics relative to all quantized phase values: inother words, the metrics associated with a given state in thevarious trellises are averaged. We will refer to this algorithmas Algorithm 1. This is the exact APP computation algorithmfor the channel with block-constant parameter described at thebeginning of Section II-B, if the observables are independent(conditionally on the parameter and the data sequence).

2) Multi-Trellis SISO Algorithm 2: Assume that the channelis slowly time-varying, i.e., we assume that ξ can exhibitsmall changes in adjacent epochs. If we allow a suitablemanipulation of {α(i)

k (sk)} and {β(i)k (sk)} only at epoch

k = lN , with l ∈ N, the possible transitions of the parameterfrom one quantization interval to another, occurring amid theblock, should be taken into account. Heuristically, we havediscovered that the impact of slow parameter changes withinthe block can be limited by performing a normalization ofthe forward state metrics {α(i)

k (sk)} (and, similarly, of thebackward state metrics {β(i)

k (sk)}) as follows:

α(i)k (sk) ←− α

(i)k (sk)∑

s′k

α(i)k (s′k)

i = 1, . . . , L ∀sk . (8)

where s′k is a dummy state in the summation, running overall Ξ states of a coherent trellis. This corresponds to anormalization of the state metrics within each FB algorithm,i.e., trellis by trellis, as opposed to a normalization amongstall trellises (as considered in Algorithm 1). We will refer tothis algorithm as Algorithm 2.

{

{{

Σ

l-th parameter block (l + 1)-th parameter block

lN − 1 lN lN + 1

ξ(1)

ξ(2)

ξ(L)

Per trellis normalization (Alg. 2)Metric sum (Alg. 1)

Fig. 2. Pictorial exemplification of the metric mixes in the two proposedalgorithms.

3) Metric Mix in the Algorithms: a Comparison: Themanipulations corresponding to (7) and (8) can be interpretedas a combining or mixing of the metrics {α(i)

k (sk)} (similarlyfor the metrics {β(i)

k (sk)}). Fig. 2 gives a pictorial descriptionof the proposed algorithmic family, highlighting the metric mixfor both Algorithms 1 and 2. Each depicted trellis diagramis associated with a coherent FB algorithm which assumes agiven channel parameter ξ(i), i = 1, . . . , L. The metric mixfor Algorithm 1 is shown to “manipulate” the metrics of alltrellises summing all metrics on a per-state basis, whereasthe metric mix for Algorithm 2 “manipulates” each trellisindependently of the other trellises, performing a per-trellisnormalization. The mix position lN , i.e., the beginning of theblock, refers to the forward metric computation. The backwardmetric computation mix is performed at epochs lN − 1.

In both Algorithms 1 and 2, the value of L, i.e., thenumber of quantized values of the channel parameter, mustbe chosen considering its impact on both performance andcomplexity. In particular, by increasing L one can improvethe performance of the proposed detection algorithms, eventhough for sufficiently large value of L the performanceimprovement becomes negligible. On the other hand, as will beshown in Section V, the complexity of the detection algorithmsincreases linearly with L.

III. DETECTION BY MULTIPLE TRELLISES FOR

PHASE-UNCERTAIN CHANNELS

In this section, the phase-uncertain channel is considered.First, the algorithms introduced in Section II are specialized tothis type of channel. Then, these algorithms are analyzed andnumerical results are given to characterize their performance.

A. SISO Detection Algorithms

In Section II-A1, the model for a channel introducing atime-invariant phase rotation θ is given. In this case, the APPof an information symbol ak is given by (2). Assuming thatθ is uniformly distributed, i.e., pθ(ϑ) = 1/2π for ϑ ∈ [0, 2π)(and 0 otherwise), expression (3) specializes to the following:

P{ak|rK0 } ∼∝ P{ak}

L∑i=1

p(rK0 |ak, ϑ(i)) (9)



where {ϑ(1), . . . , ϑ(L)} is a set of L properly chosen phasevalues [28]. This detection approach for channels with a block-constant random phase was used in [8].

If we assume a slowly varying channel phase (i.e., the band-width of the channel parameter process is small compared withthe receiver filter bandwidth), the discrete-time observable canbe modeled as in (4) by incorporating a time-varying phaseprocess {θk}:

rk = ckejθk + nk . (10)

where |ck| = 1 (DE-QPSK is considered) and nk is a discrete-time complex AWGN process with Var{nk} = (REb/N0)−1,in which R is the system spectral efficiency in bits per channeluse. By suitably modeling the stochastic process {θk}, onecould try to develop an exact APP algorithm. Since we do notwant to rely on exact channel parameter statistics, we resortto the multi-trellis SISO algorithms described in Section II-B.

B. Numerical Results

In this section, we assume that transmission over an AWGNchannel is affected by a Wiener phase noise process {θk}described by the following recursive relation:

θk = θk−1 + wk mod 2π (11)

where {wk} is a sequence of i.i.d zero mean Gaussianvariables. The standard deviation of wk, denoted as σθ , isrepresentative of the phase noise intensity. As mentioned inSection I, the chosen modulation format is DE-QPSK.

Both Algorithms 1 and 2 introduced in Section III areconsidered. For Algorithm 1, the number of quantized phasevalues is L = 32, i.e., 8 values per phase interval betweenadjacent QPSK symbols: in other words ϑ(i) = 2πi/32,i = 0, . . . , 31. For Algorithm 2, L = 8 and ϑ(i) = 2πi/32,i = 0, . . . , 7, accounting for a phase interval of only π/2. Infact, it is well known that for a time-invariant channel phase,the symmetry of DE-QPSK enables to perform detectionaccounting only for a phase interval (0, π/2) [8], [21]. Theparticular structure of Algorithm 2, i.e., the fact that thenormalization is carried out on a “per-coherent trellis” basis,allows to perform this complexity reduction at the cost of alimited penalty also in the presence of a time-varying channelphase.

In Fig. 3, Algorithms 1 and 2 are investigated for detec-tion of DE-QPSK without an outer code. The informationsymbols are transmitted in blocks. The bit error rate (BER)performance relative to the first 120 bits in the transmittedblocks is shown, as a function of the bit position within theblock. The bit SNR Eb/N0 is equal to 6 dB. The inter-mixinterval is N = 15 and the phase noise parameter σθ = 5◦.One can note that the curves exhibit periodicity 2N , since eachQPSK symbol encodes 2 bits. In particular, for both algorithmsthe BER is lowest at the epoch in the middle between twoconsecutive metric mixes. Algorithm 1, moreover, exhibits afloor, since bits corresponding to the metric mix epochs, i.e.,bits at positions l2N and l2N + 1, are randomly decided.This floor has little impact in a concatenated coded scheme, asconsidered in the following paragraphs, since bits at positionsl2N and l2N + 1 are characterized by APP equal to 0.5, i.e.,they behave as punctured bits.

Algorithm 1Algorithm 2

10−3

0 20 40 60 80 100 120Bit Position

BE

R

10−1

1

10−2

Fig. 3. BER performance at each codeword position for Algorithms 1 and 2and DE-QPSK. N = 15 and Eb/N0 = 6.0 dB.

We now assume that the information sequence is encodedby an outer regular (3,6) LDPC code [26]. The codewordlength is set to 6000 bits. The decoder uses a standard LDPCdecoder as an external SISO module which exchanges extrin-sic information with the DE-QPSK inner detector, where theproposed SISO algorithms are used instead of a coherent SISOalgorithm for DE-QPSK. This can obviously be interpreted asa serially concatenated coding scheme. More details on theconsidered concatenated system structure can be found in [28].The maximum number of iterations is set to 100.

In Fig. 4, the performance of the described schemes isshown in terms of BER versus SNR. The performance fortransmission over an AWGN channel without phase noise,considering an ideal coherent FB algorithm as inner detec-tor, is shown as a reference. The remaining curves showthe performance obtained with the proposed algorithms. Inparticular, the curves marked as “Alg1” and “Alg2” correspondto the performance of the schemes with Algorithms 1 and 2,respectively. For each algorithm, several values of the phasenoise standard deviation σθ (given in degrees in the figurelegend) are considered. In each case, the inter-mix intervalN is heuristically optimized. The results in Fig. 4 show that,even in the presence of a significant phase noise (for instance,σθ = 10◦), it is possible to “blindly” process the metrics ofthe trellises while still achieving an SNR loss as limited as1 dB. Heuristically, the optimum value of N turns out tobe inversely proportional to σθ . The results in Fig. 4 showthat Algorithm 2 entails better performance than Algorithm 1.In particular, for very strong phase noise, i.e., σθ = 10◦,Algorithm 1 suffers an SNR penalty larger than 1 dB withrespect to Algorithm 2. This is due to the fact that Algorithm 1completely erases the phase information every N time steps,whereas Algorithm 2 performs only a “trellis balancing” asdescribed in Section II-B2.

In Fig. 5, a direct comparison between the performance (interms of BER as a function of the SNR) with Algorithm 1and Algorithm 2, for a fixed value of the inter-mix distanceN = 15, and several values of σθ , is shown. The value N = 15optimized the system performance at σθ = 5◦, as shownin Fig. 4. The remaining system and simulation parameters



Alg. 2, σθ = 3◦, N = 30Alg. 2, σθ = 5◦, N = 15Alg. 2, σθ = 10◦, N = 2

Alg. 2, σθ = 0◦, N = 200

Alg. 1, σθ = 0◦, N = 200Alg. 1, σθ = 3◦, N = 20Alg. 1, σθ = 5◦, N = 15

Alg. 1, σθ = 10◦, N = 10

1.5 2 2.5 3 3.5 4 4.5 5

10−3

1

10−1

10−2

10−4

10−5

BE

R

Eb/N0

Coherent DE-QPSK

Fig. 4. BER performance of LDPC-coded DE-QPSK schemes where theproposed algorithms (both Algorithms 1 and 2) are used.

are those of Fig. 4. The BER curves show clearly that forvalues of the phase noise parameter σθ lower than or equal to5◦, decoding convergence is guaranteed for approximately thesame SNR, whereas if σθ > 5◦ convergence is not guaranteedany longer, i.e., an error floor may appear. In particular, theerror floor characterizing the BER curve corresponding toAlgorithm 2 with σθ = 10◦ is due to the fact that, in order tocope with a strong phase noise, Algorithm 2 needs a very smallinter-mix interval N , as clearly shown in Fig. 4. From theresults in Fig. 5, one can conclude that the proposed algorithmsare blind with respect to the phase noise intensity as long asthis intensity is lower than that considered in the algorithmdesign.

In Fig. 6, the SNR needed to achieve a BER equal to 10−3

is shown as a function of σθ for both considered algorithms.The system and the simulation parameters are those of Fig. 4.One can conclude that the proposed algorithms are blind withrespect to the phase noise intensity σθ , as long as this intensityis lower than a particular value which is a function of N .Beyond this critical value, the SNR needed to achieve thegiven BER value, i.e., 10−3, diverges rapidly.

From the results in Figures 4, 5 and 6, one can concludethat, in the considered phase-uncertain channel scenario, Al-gorithm 2 performs better than Algorithm 1. This can beattributed to the strong approximations made by Algorithm 1in “erasing” the phase memory at regular intervals, in an en-vironment in which the phase varies slowly, yet continuously.This leads to wrong metrics in the proximity of mix epochs,where the erase operation is carried out. In other words,Algorithm 1 periodically enforces the strongest metrics among

2 2.5 3 3.5 4 4.5 5

Alg.2 σθ =10◦

Alg.1 σθ =10◦

Alg.2 σθ =5◦Alg.2 σθ =3◦Alg.2 σθ =2◦Alg.2 σθ =1◦

Alg.1 σθ =5◦Alg.1 σθ =3◦Alg.1 σθ =2◦Alg.1 σθ =1◦

10−5

10−4

10−3

10−2

10−1

100

BE

R

Eb/N0

Fig. 5. BER performance, as a function of the SNR, of the proposedAlgorithms 1 and 2. Several values of σθ are considered and N = 15.

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14

Alg. 2 N = 100

Alg. 1 N = 100Alg. 1 N = 10Alg. 1 N = 15Alg. 1 N = 3

Alg. 2 N = 10Alg. 2 N = 15Alg. 2 N = 3

Eb/N

0

σθ

Fig. 6. SNR needed to achieve BER= 10−3, as a function of σθ , consideringAlgorithms 1 and 2.

the trellises, whereas Algorithm 2 keeps the distribution ofthe metrics inside each trellis but periodically erases thedifferent weightings of the trellises. This enables Algorithm 2to account for channel phase variations during an N -symbolblock.

IV. DETECTION BY MULTIPLE TRELLISES FOR FLAT

FADING CHANNELS

In this section, the flat fading channel is considered. First,we derive the FB algorithm assuming a Markov chain model



for the fading channel. Then, we specialize the algorithmintroduced in Section II to the case of flat fading channel, high-lighting its similarities with the Markov chain-based approach.Finally, the algorithms are analyzed and their performance ischaracterized through numerical results.

A. FB Algorithm for Fading Channels based on MarkovChains

The time-invariant flat fading model given in (5) can be ex-tended to a more realistic model with time-varying flat fading.Accordingly, the discrete-time observable can be expressed as

rk = fk ck + nk (12)

where {fk} is the fading process.5 In the presence of Rayleighfading, each fading realization fk can be modeled as azero-mean complex circularly symmetric Gaussian RV. Weassume that the fading process {fk} is modeled accordingto Clarke [15], [16], with zero mean, unit variance andautocorrelation function Rf (n) = J0(2πnfDT ), where J0(·)is the zero-th order Bessel function and fDT is the maximumnormalized Doppler shift which characterizes the speed of thefading process.

We now outline the derivation of a simple first-orderMarkov chain model which approximately describes the evo-lution of the complex fading process. Several papers dealwith Markov-chain modeling of the fading process—for moredetails, we refer the reader to [18], [29], [30] and referencestherein. We first partition the complex plane into Nphase angu-lar sectors [2π i−1

Nphase, 2π i

Nphase), i = 1, . . . , Nphase. Then, we

further split each sector into Nampl “ring-shaped” regions. Asa consequence, the complex plane is split into NphaseNampl

sub-domains {Dij} where Dij denotes the domain corre-sponding to the i-th phase sector and the j-th ring-shapedregion. In Fig. 7, an illustrative example with Nphase = 8angular sectors and Nampl = 2 ring-shaped regions is shown.

By associating the fading regions with states, it is possible todescribe the evolution of the fading process through the use ofa Markov chain. In general, considering a first-order Markovmodeling for the fading process,6 the total number of fadingstates is L = NamplNphase. The probabilities of transitionthrough different fading states can be computed through propernumerical integrations. For example, in order to evaluate theprobability of transition from the region Dij to the regionDkl, one can follow the method in [29], which is accurate aslong as the first-order Markov chain modeling of the fadingprocess holds and, in turns, corresponds to a scenario wherethe fading process is sufficiently slow [18].

Since the fading process is modeled through a Markov chainwhose state corresponds to the current fading subregion Dij , itis possible to derive a proper FB algorithm for the computationof the APPs of the transmitted symbols {ak}. A generalformulation accounting for a finite-memory channel depending

5We remark that this discrete-time model can be obtained from thecontinuous-time multiplicative fading model assuming that the fading processhas a bandwidth much smaller than the signal bandwidth.

6We remark that the considered approach can easily be extended to higher-order Markov models of the fading process, at the expense of an increasednumber of fading states.

D11

D21

D22D32

D42 D31

D41

D51

D52 D61 D71

D81

D82

D72D62

D12

Fig. 7. Partitioning of the fading complex plane into fading regions.

on a generic process {ξk} modeled by a Markov chain canbe found in the Appendix. In particular, the FB algorithm inthe Appendix assumes a channel whose output observablesare independent conditionally on the data sequence and theparameter sequence. For instance, this is the case for thefading model in (12), where the observables are conditionallyindependent and Gaussian.

In the following, we will assume that the symbols {ak} arequaternary and encoded by a DE-QPSK encoder before trans-mission. The channel parameter ξk corresponds to the fadingregion f̂k ∈ {Dij} i = 1, . . . , Nphase, j = 1, . . . , Nampl.The extended state described in the Appendix here is σk =(sk, f̃k), where sk is the DE-QPSK encoder state at epoch k,and the fading region f̃k has been substituted to the genericparameter ξk.

The two essential ingredients needed for actual implemen-tation of the Markov chain-based SISO algorithm in a scenariowith fading are the transition probability P{f̃k+1|f̃k} betweenthe Markov chain states f̃k and f̃k+1, obtained by suitablymodeling the fading Markov chain, and the conditional PDFof the observable p(rk|ak, f̃k, sk), given by the following

p(rk|ak, f̃k, sk) =p(rk, f̃k|ak, sk)

p{f̃k}

=

∫f̃k

p(rk|f, ak, sk)pf (f)df∫f̃k

pf (f)df

(13)

where the independence between the fading process and theDE-QPSK coded data sequence ck is exploited, p(rk|f, ak, σk)is a Gaussian PDF (with mean fck), and pf (f) is the PDF ofthe fading coefficient.

B. Multi-Trellis SISO Algorithm for Fading Channel

The concept of detection by multiple trellises can be nowdirectly applied to a fading channel. In particular, as for the



phase-uncertain channel, if the channel is characterized byblock-constant fading, Algorithm 1 is an optimum solution.In order to simplify the metric computation, the integral in(13) will be approximated by a finite sum of simple Gaussianmetrics. We observed that this can lead to numerical problemsat high SNR, where the noise variance becomes small. Toovercome this problem, one may increase the accuracy ofthe numerical integration techniques used to compute (13),or prevent the variances of the Gaussian pdfs to become toosmall and trigger numerical problems.

Observe that every concatenated scheme with a powerfulerror correction code is characterized by a bad BER perfor-mance below a given SNR threshold and an operational BERperformance beyond this threshold.7 If the detection algorithmassumes a given, fixed, SNR value, one is guaranteed to obtainthe performance of the same detection algorithm using thecorrect SNR value only when the actual SNR value and thefixed one are equal. Even with fixed SNR value assumed bythe detection algorithm the BER as a function of the SNRis still expected to be monotonically decreasing. Therefore, ifthe assumed SNR is fixed to guarantee an operational BER atthat very SNR value, the fixed SNR algorithm will guaranteeoperational BER beyond this SNR as well. As a consequence,we chose to fix the variance of the Gaussian metric, i.e.,the SNR assumed by the detection algorithm, and to makeit independent of the actual noise variance. This allows toovercome numerical problems and leads to a completely blinddetection algorithm, which does not need either knowledge offading or noise statistics.

C. Numerical Results

Unlike several works in the literature, where the fadingprocess used in the simulations is generated according to theconsidered Markov-chain model, in the following the fadingprocess used in the simulations is generated according to arealistic Clarke model.

In order to verify the effectiveness of the proposed detectionby multiple trellises approach, we consider its applicationto the cases with uncoded DE-QPSK and with a regular(3,6) LDPC code with codeword length 32000—this lengthis necessary in order to combat long fades. The code should,in fact, “observe” a received sequence long enough to accu-rately describe the statistics of the channel, i.e., to exploit itsergodicity. We performed simulations considering Nampl = 2and Nphase = 16 and considering Algorithm 1 and theproposed simplified metric scheme. Algorithm 2, in the caseof fading channel, exhibits unacceptable performance, and,therefore, is not shown. This is due to the fact that the mixoperation in Algorithm 2 assigns large weights to trellisescharacterized by incorrect fading amplitudes. The considerednormalized Doppler rate fDT is equal to 0.01, correspondingto a moderately fast fading channel. The obtained results areshown in Fig. 8. The multi-trellis curve is obtained assuminga noise variance value corresponding to an SNR of about7 dB. The inter-mix interval is heuristically optimized by trial

7In actual systems, the transition from bad BER performance to operationalBER is not perfectly sharp, i.e., it happens within a small SNR region, usuallyreferred to as waterfall region.

0 5 10 15 20E

b/N

0

10-5

10-4

10-3

10-2

10-1

100

BE

R

BPSK with perfect CSIMarkov Chain FB N

phase=16 N

ampl=2

Multi-trellis Nphase

=16 Nampl

=2 Nmix

=20

Perfect CSI

Fig. 8. BER performance, as a function of the SNR, in a scenario witha flat Rayleigh fading channel. Various schemes are considered: (i) BPSKwith perfect CSI, (ii) LDPC-coded QPSK with Markov chain-based FB, (iii)an LDPC-coded QPSK with multi-trellis SISO, and (iv) LDPC-coded QPSKwith perfect CSI.

and error and set to 20. In every LDPC-coded scheme, 30decoding iterations are considered at the receiver side. TheMarkov chain-based algorithm presented in Section IV-A isalso investigated and its performance is shown. As a reference,the performance of (i) the described concatenated scheme and(ii) an uncoded BPSK signaling, both considering perfect CSI,is also shown. As one can immediately see, the performanceloss incurred by the use of the proposed detection by multipletrellises can be quantified at about 1 dB in comparison withthe Markov-chain model performance and 1.8 dB comparedwith the perfect CSI scenario.

V. COMPLEXITY ANALYSIS AND DISCUSSION

In this section, we investigate the complexity of the pro-posed multi-trellis SISO algorithms with a simple-minded, yetmeaningful, approach. In order to highlight the advantages ofthe proposed algorithms, we compare their complexities withthat of the general finite-memory approach to detection forchannels affected by uncertain parameters described in [9] andwith the Markov chain-based approach. We will evaluate thecomputational complexity in terms of elementary operations(i.e., additions or multiplications) per trellis section during asingle recursion.

We preliminarily denote as Compcoher the complexity of anFB algorithm used by a coherent detector. It is possible toshow that this complexity is

Compcoher = Θ(ΞM)

where M is the cardinality of the information symbol set andthe notation Θ(·) stands for “on the order of.” For simplicity,we assume that the complexity of the coherent receiver is thesame in terms of multiplications and additions. Moreover, for



ease of comparison between different algorithms, we assumethat Compcoher is exactly ΞM .

We first evaluate the complexity of the proposed multi-trellisSISO algorithms, namely Algorithm 1, described in Subsec-tion II-B1, and Algorithm 2, described in Subsection II-B2.For both algorithms, L trellis diagrams (each one equal tothat of the coherent FB algorithm) are used. Therefore, thisincreases the complexity of the proposed SISO algorithms toL Compcoher. At this point, one has to consider the additionalcomplexity of the mix operations.

• For Algorithm 1, from the updating rule (7) one canconclude that 1 addition (over L quantized values of thechannel parameter, i.e., phase or fading) for each state hasto be carried out. Therefore, ΞL supplementary additionshave to be considered. Since a mix operation takes placeevery N transitions, the complexity increase, in terms ofelementary operations per trellis section, is ΞL/N .

• For Algorithm 2, from the updating rule (8) one canconclude that 1 addition (over Ξ states) for each com-ponent trellis diagram has to be performed. At thispoint, one division has to be carried out per state andtrellis component. Therefore, a mix operation requires,overall, LΞ additions and LΞ divisions. The complexityincrease, per trellis section, is therefore, 2LΞ/N in termsof elementary operations.

The finite-memory FB algorithm described in [9] is char-acterized by a “trellis expansion,” in order to partially takeinto account the channel memory. This memory expansion isdescribed by a finite-memory parameter Nfm, which character-izes the number of additional information symbols consideredin the definition of a state in the trellis diagram at the receiver.The number of states for the computation of the state metricsin a detector/decoder, where a finite-memory FB algorithmis used, is ΞMNfm . Therefore, one can conclude that thecomplexity increases proportionally to MNfm . Denoting byCompfm the complexity per trellis section (either in terms ofadditions or multiplications/divisions) in each recursion of afinite-memory FB algorithm, one can write:

Compfm = CompcoherMNfm = ΞMNfm+1.

The complexity of the first-order Markov chain model [18]analyzed in Section IV is proportional to the number of statesof the coherent FB algorithm Ξ, to the cardinality of the inputsymbol space M , and to the square of the number of quantizedvalues of the channel parameter, i.e., L2. In other words, thecomplexity of a Markov chain-based FB algorithm is given by

CompMC = CompcoherL2 = ΞML2.

The complexity of the proposed multi-trellis SISO al-gorithms, the finite-memory FB algorithm in [9], and theMarkov chain-based FB algorithm are summarized in Table I.We remark that the complexity computation is, in practice,implementation-dependent. A noteworthy case is the imple-mentation of the considered algorithms on a generic purposeprocessing unit, which usually leads to the serial computationof the quantities involved in the FB algorithms. In particular, itis well known that, due to the negative exponential behavior of

TABLE ICOMPLEXITY PER TRELLIS SECTION, DURING A SINGLE RECURSION, OF

VARIOUS ALGORITHMS.

Algorithm Complexity

SISO Algorithm 1 ΞML +ΞL

N

SISO Algorithm 2 ΞML +2ΞL

N

Finite-memory ΞMNfm+1

Markov Chain ΞML2

the forward and backward state metrics, periodic normaliza-tion of these metrics, carried over all states and all trellises,is needed. This normalization takes place at arbitrary timeepochs, but usually every 10–100 time steps. This normaliza-tion could be easily modified in order to implement the metricmixes described in this paper.

Moreover, the proposed multi-trellis SISO algorithms areintrinsically highly parallelizable. Exploiting properly thischaracteristic in the implementation could lead to significantincrease of the decoding throughput, as desirable in futurehigh data-rate wireless communication systems.

At this point, a careful reader can observe that since thefinite-memory approach in [9] is very different from the multi-trellis SISO algorithms proposed in this paper, a meaningfulcomplexity comparison between these algorithms should becarried out for a given performance level. To this purpose,we consider a complexity comparison for the same BERperformance at the same SNR, using the same LDPC-codedDE-QPSK scheme used in Section III-B, in a scenario withphase noise. As shown in Fig. 9, the performance obtainedby Algorithm 2 with N = 15 and L = 8, in a phasenoise scenario with σθ = 5◦, is approximately approachedby the finite-memory FB algorithm with Nfm = 3 (within asmall fraction of a dB). The remaining simulation parametersare set as in Section III-B. The complexity of Algorithm 2is ΞML + 2LΞ/N � 32Ξ, whereas the complexity of thefinite-memory FB algorithm is ΞM3+1 = 256Ξ. This largedifference is due to the fact that the complexity of the finite-memory FB algorithm grows exponentially with the memorylength (quantified by Nfm). This exponential increase of thecomplexity can be seen as a paradox characterizing the finite-memory approach, since slower phase processes require largervalues of Nfm. As a consequence, good channels, exhibitingslow phase variations, need larger complexity than bad chan-nels, with high phase dynamics. The proposed multi-trellisSISO algorithms are a pragmatic solution to overcome thisparadox, since high values of N require lower complexity atthe receiver.

In [8], [31], [32], other low complexity approaches tocombat phase noise impairment are proposed, on the basis ofa block-constant phase assumption. The additional strength ofour multi-trellis SISO algorithms consists of their capability totake into account possible parameter changes within a block,in conjunction with the complete blindness with respect to thechannel statistics.



1

10−1

10−2

10−3

10−4

10−5

BE

R

2 2.5 3 3.5 4 4.5 5Eb/N0

Alg.2 N = 15Alg. f.m. Nfm = 3

Alg.1 N = 15Alg. f.m. Nfm = 2

Fig. 9. BER performance of LDPC-coded DE-QPSK schemes whereAlgorithm 2 and a finite-memory (f.m.) detection algorithm are used. Thephase noise parameter is σθ = 5◦.

VI. CONCLUSIONS

In this paper, we have introduced a novel approach, referredto as detection by multiple trellises, suitable for transmissionover channels affected by time-varying parameters. Afterintroducing the basic idea in an intuitive way, we haveconsidered two relevant applications: detection for phase-uncertain and fading communications. The idea of the pro-posed approach consists of using several parallel trellises,over which coherent FB algorithms, each associated with aproper quantized value of the stochastic parameter, run. Inorder to cope with time-varying processes, the forward andbackward metrics in the parallel FB algorithms are properly“mixed” together at regular intervals. For a scenario with phasenoise, two multi-trellis SISO algorithms have been proposed,considering uncoded and LDPC-coded DE-QPSK transmis-sion over an AWGN channel with Wiener phase noise. In thescenario with fading, after deriving an FB algorithm basedon a simple first-order Markov chain model for the fadingprocess, we have considered its multi-trellis extension. In allcases, DE-QPSK has been the used modulation format. Aninteresting feature of the proposed algorithms is the fact thatthey do not require knowledge of the statistics of the stochasticparameter (either the phase or fading), i.e., they are blind.Given their low complexity and high parallelizability, theproposed multi-trellis SISO algorithms are attractive for futurehigh-throughput wireless communication systems. While thederivations have been carried out for FB algorithms [25], theproposed approach extends directly to trellis-based sequencedetection algorithms, such as the Viterbi algorithm [33].

APPENDIX

In this appendix, an extension of the standard FB algorithmto a channel whose statistics at epoch k are a function of thestate ξk of a Markov chain is described. Let us assume that,given {ξk}, the modulator-channel pair can be described by anFSM, in the sense that the observable statistics are functions

of the state σk of an FSM whose input is the informationsymbol sequence {ak}. Moreover, let us assume that (i) {ak}and {ξk} are independent and (ii), given {ak} and {ξk}, theobservables are independent. Following the guidelines in [9],[20], [21], it can be shown that the a posteriori probability ofthe symbol ak can be computed as follows:

P{ak|rK0 } =

∑(σk,σk+1):ak

βk+1(σk+1)αk(σk) γk(σk, σk+1, ak)

(14)where, as before, rK

0 denotes the vector of the observablesand σk = (sk, ξk) is the (extended) state of the system; thenotation (σk, σk+1) : ak denotes “the set of all (σk, σk+1)pairs compatible with the input symbol ak” and the branchmetric γk(σk, σk+1, ak) is defined as

γk(σk, σk+1, ak) = p(rk|ak, ξk, sk) · P{ak} · P{ξk+1|ξk}(15)

in which P{ξk+1|ξk} is the transition probability between theMarkov chain states ξk and ξk+1, and p(rk|ak, ξk, sk) is thechannel statistical description, i.e., the observable PDF giventhe data sequence and the channel parameter ξk. The forwardand backward metrics αk(σk) and βk(σk) are obtained withthe following recursions:

αk(σk) =∑

(σk−1,ak−1):σk

αk−1(σk−1) γk−1(σk−1, σk, ak−1)

βk(σk) =∑

(σk+1,ak):σk

βk+1(σk+1) γk(σk, σk+1, ak) .

The FB algorithm in (14) operates on a trellis whose number ofstates is the number Ξ of states of the modulator-channel FSMtimes the number L of states of the channel parameter Markovchain. This can be interpreted as a “super-trellis” comprisingL trellises, each with Ξ states.

As special case, if the Markov chain {ξk} is time-varyingand the transition matrix differs from the identity matrixonly at time epochs k = Nl, with l ∈ N, it can be easilyshown that the forward and backward recursions in the aboveextended FB algorithm are equivalent to the computation ofL independent forward and backward recursions in the Ξ-state trellises for N − 1 time steps. Every N time steps, therecursions involve, in general, all trellises. This correspondsto a block-constant discrete parameter ξk, which has beendiscussed in Section II-B assuming uniform distribution ofthe parameter realization. The corresponding super-trellis isshown in Fig. 1.

ACKNOWLEDGMENT

The authors wish to thank ing. Aldo Curtoni for his contri-butions to the simulation results.

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Michele Franceschini was born in Milan, Italy, in1977. He received the Dr. Ing. degree ("Laurea,"5-year program) in Electrical Engineering (summacum laude) from the University of Parma in 2002.In 2003, he received the "Paolo Conti Award" asthe best graduate in Information Engineering at theUniversity of Parma in the academic year 2002. In2006, he received the Ph.D. from the Universityof Parma. From 2006 to 2008 he held a postdocposition at the University of Parma. Since March2008, he holds a postdoc position at the IBM T.J.

Watson Research Center, Yorktown Heights, NY. His research interests lie inthe area of communication and information theory, storage channels, low-density parity-check code design, theoretical aspects of optical communi-cation, advanced signal processing techniques, synchronization, and low-complexity implementation of digital communication systems.

Gianluigi Ferrari was born in Parma, Italy, inNovember 1974. He received the “Laurea" degree(five-year program) (summa cum laude) in electricalengineering and the Ph.D. degree in InformationTechnologies from the University of Parma in Oc-tober 1998 and January 2002, respectively. FromJuly 2000 to December 2001, he was a VisitingScholar at the Communication Sciences Institute,University of Southern California, Los Angeles, CA,USA. Since 2002, he has been a Research Professorwith the Department of Information Engineering,

University of Parma, where he is now the coordinator of the Wireless Ad-hoc and Sensor Networks (WASN) Laboratory. Between 2002 and 2004, hevisited several times, as a Research Associate, the Electrical and ComputerEngineering Department at Carnegie Mellon University, Pittsburgh, PA. In fall2007 he visited, as a DUO-Thailand Fellow, the King Mongkut’s Institute ofTechnology Ladkrabang (KMITL), Bangkok, Thailand.

Dr. Ferrari has published more than 100 papers in leading internationalconferences and journals. He is coauthor of the books Detection Algorithmsfor Wireless Communications, with Applications to Wired and Storage Systems(Wiley: 2004), Introduzione a Teoria della probabilità e variabili aleatoriecon applicazioni all’ingegneria e alle scienze, (Editrice Esculapio-ProgettoLeonardo:2008), Ad Hoc Wireless Networks: A Communication-Theoretic Per-spective (Wiley: 2006), and LDPC Coded Modulations (Springer: 2009). Hisresearch interests include digital communication systems analysis and design,wireless ad hoc and sensor networking, adaptive digital signal processing, andinformation theory.

Dr. Ferrari is a co-recipient of a best student paper award at the 2006International Workshop on Wireless Ad hoc Networks (IWWAN’06). He actsas a frequent reviewer for many international journals and conferences. Heacts also as a technical program member for many international conferences.He currently serves on the Editorial Boards of THE OPEN ELECTRICAL

AND ELECTRONIC ENGINEERING (TOEEJ) JOURNAL (Bentham Publishers),the INTERNATIONAL JOURNAL OF RF TECHNOLOGIES: RESEARCH AND

APPLICATIONS (Taylor & Francis), and the INTERNATIONAL JOURNAL

OF FUTURE GENERATION COMMUNICATION AND NETWORKING (SERSC:Science & Engineering Research Support Center).



Riccardo Raheli received the Dr. Ing. (Laurea) de-gree in Electrical Engineering “summa cum laude"from the University of Pisa, Italy, in 1983, theMaster of Science (M.S.) degree in Electrical andComputer Engineering from the University of Mas-sachusetts at Amherst, U.S.A., in 1986, and theDoctoral (Perfezionamento) degree in Electrical En-gineering “summa cum laude" from the ScuolaSuperiore S. Anna di Studi Universitari e di Per-fezionamento, Pisa, Italy, in 1987.

From 1986 to 1988 he was with Siemens Teleco-municazioni, Milan, Italy. From 1988 to 1991, he was a Research Professorat the Scuola Superiore S. Anna, Pisa, Italy. In 1990, he was a VisitingAssistant Professor at the University of Southern California, Los Angeles,U.S.A.. Since 1991, he has been with the University of Parma, Italy, first as aResearch Professor, then Associate Professor and currently Professor of Com-munications Engineering and Chairman of the Communications EngineeringProgram Committee. His scientific interests are in the general area of statis-tical communication theory, with application to wireless, wired and storagesystems, and special attention to data detection in uncertain environments,iterative information processing and adaptive algorithms for communications.His research work has lead to numerous scientific publications in leadinginternational journals and conference proceedings, as well as a few industrialpatents. In 1990, he conceived (with Andreas Polydoros) the principle of “Per-

Survivor Processing." He is coauthor of a few scientific monographs suchas Detection Algorithms for Wireless Communications, with Applications toWired and Storage Systems (John Wiley & Sons, 2004) and LDPC CodedModulations (Springer, 2009). He is coauthor of the paper which receivedthe “2006 Best Student Paper Award in Signal Processing & Coding for DataStorage" from the IEEE Communications Society.

Dr. Raheli served on the Editorial Board of the IEEE TRANSACTIONS ONCOMMUNICATIONS, as an Editor for Detection, Equalization and Coding,from 1999 to 2003. He was the leading Guest Editor of a special issue ofthe IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS (JSAC)on “Differential and Noncoherent Wireless Communications," published in2005. He served on the Editorial Board of the “European Transactions onTelecommunications (ETT)," as an Editor for Communication Theory, from2003 to 2008. He has also served on the Technical Program Committeeof many leading international conferences in the areas of CommunicationTheory and Systems, Information Theory and Signal Processing, such asthe IEEE International Conference on Communications (ICC), IEEE GlobalCommunications Conference (GLOBECOM), European Signal ProcessingConference (EUSIPCO), IEEE International Symposium on Power-LineCommunications and Its Applications (ISPLC), International Symposium onInformation Theory and its Applications (ISITA), International Symposiumon Spread Spectrum Techniques and Applications (ISSSTA), and others.


Detection by multiple trellises

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